91Ó°ÊÓ

The substitution method is a technique used in calculus to simplify the process of evaluating integrals. It works by replacing a complex expression within the integral with a single variable. This is especially useful when dealing with integrals that are difficult to solve directly.

For instance, when we face an integral like \( \int 2x \cos(\pi x^2) \, dx \), using substitution can make the integral much more manageable. In this specific exercise, we set \( u = \pi x^2 \). This substitution allows us to express the differential \( dx \) in terms of \( du \):

• Differentiate \( u \) with respect to \( x \): \( du = 2\pi x \, dx \).
• Solve for \( x \, dx \): \( x \, dx = \frac{du}{2\pi} \).

By substituting \( u \) back into the integral, we transition from integrating with respect to \( x \) to integrating with respect to \( u \), resulting in \( \int \frac{1}{\pi} \cos(u) \, du \). This transformation simplifies the calculation, making the integration process smoother and more straightforward.

A definite integral computes the accumulation of quantities, such as areas under curves, between two bounds or limits. Unlike an indefinite integral, which gives a function, a definite integral results in a specific numerical value.

In this exercise, we are looking at the integral \( \int_{1/2}^{1} 2x \cos(\pi x^2) \, dx \), which is a classic example of a definite integral. Here, the bounds are from \( x = 1/2 \) to \( x = 1 \). During substitution, these bounds are also transformed according to the substitution rule.

When substitution \( u = \pi x^2 \) is applied, the limits of \( u \) change accordingly:

• When \( x = 1/2 \), \( u = \pi (1/2)^2 = \pi/4 \).
• When \( x = 1 \), \( u = \pi \).

The definite integral then computes the value of \( \int_{\pi/4}^{\pi} \frac{1}{\pi} \cos(u) \, du \), which after solving, provides the precise numerical output \(-\frac{\sqrt{2}}{2\pi}\). This result represents the area under the transformed function within the specified limits.

The cosine function, denoted as \( \cos(x) \), is a fundamental trigonometric function representing the x-coordinate of a unit circle as a function of an angle \( x \). Its properties include being an even function, periodic with a period of \( 2\pi \), and oscillating between -1 and 1.

In the given exercise, the function of interest is \( \cos(\pi x^2) \). This translates the typical behavior of the cosine function into a more complex argument \( \pi x^2 \), which shifts the timing of the oscillations. Despite this complex appearance, the original properties of the cosine still apply.

Key aspects of the cosine function to remember include:

• It is periodic with identical values repeating every \( 2\pi \) units.
• It is even, meaning \( \cos(-x) = \cos(x) \).
• The integral of \( \cos(u) \) over a range can be calculated efficiently since \( \int \cos(u) \, du = \sin(u) \).

Understanding these characteristics helps simplify the handling of trigonometric functions in integrals, making them crucial tools in calculus when dealing with functions like \( \cos(\pi x^2) \) as seen in this problem.